B Can an Object with N Dimensions Exist in N-1 Dimensions?

  • B
  • Thread starter Thread starter duyix
  • Start date Start date
  • Tags Tags
    2d 3d Space
duyix
Messages
1
Reaction score
1
I am concerned that this question may instead be a philosophical one although if it it mathematical, any insights would be very appreciated. The question is this; could an object of N dimensions exist entirely in N-1 dimensions? In other words, could an infinitely flat object have 3 degrees of freedom and also be able to fit entirely in 2D space? Thank you and please excuse any naivety
 
Mathematics news on Phys.org
duyix said:
The question is this; could an object of N dimensions exist entirely in N-1 dimensions?
No, it's not possible.

duyix said:
In other words, could an infinitely flat object have 3 degrees of freedom and also be able to fit entirely in 2D space? [\quote]
If by "infinitely flat object" you mean "a plane" it's already a two-dimensional object that can be determined by two nonparallel direction vectors. I.e., two degrees of freedom.
 
Last edited:
There are different definitions of the term Dimension. One of them is that of number of data points needed to fully describe every point in the n-th dimensional object. And that number is precisely n.
There are results to the effect that ##\mathbb R^{n+k} ; k >0 ##; k a positive Integer, cannot be embedded in ##\mathbb R^n ##. There are similar results for n-spheres ## S^n ##. that cannot be embedded in ## \mathbb R^n ## or lower IIRC, the main result is that of Borsuk -Ulam.

Edit: A 1-dimensional object embedded in n-space is describable as ##(f_1(x), f_2(x),...,f_n(x))##.
An m-dimensional object in k-space is describable as ## (f_1(x_1,..., x_m), f_2(x_1,x_2,..,x_m),,..,f_k(x_1,x_2,..,x_m) )##
 
Last edited:
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.

Similar threads

Back
Top